GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 11 Jul 2020, 20:01

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If x is a positive integer, how many values of x satisfy the..........

Author Message
TAGS:

### Hide Tags

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3410
If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

31 Oct 2018, 03:28
1
21
00:00

Difficulty:

95% (hard)

Question Stats:

35% (02:03) correct 65% (02:20) wrong based on 281 sessions

### HideShow timer Statistics

If x is a positive integer, how many values of x satisfy the inequality, $$\frac{(x - 2)}{(x^2 – 13x + 36)} ≤ 0$$?

A. 4
B. 5
C. 6
D. 7
E. Cannot be determined

_________________
Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 950
WE: Supply Chain Management (Energy and Utilities)
Re: If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

31 Oct 2018, 08:56
1
1
1
EgmatQuantExpert wrote:
If x is a positive integer, how many values of x satisfy the inequality, $$\frac{(x - 2)}{(x^2 – 13x + 36)} ≤ 0$$?

A. 4
B. 5
C. 6
D. 7
E. Cannot be determined

$$\frac{(x - 2)}{(x^2 – 13x + 36)} ≤ 0$$

Factorizing denominator,

Or, $$\frac{x-2}{\left(x-4\right)\left(x-9\right)}\le \:0$$

Now, applying Wavy-curve method, we have the following ranges of x:-

$$x\leq{2}$$ or 4<x<9

As 'x' is a positive integer, hence the posssible value of x:- 1,2,5,6,7,8

Therefore, SIX nos. of values of 'x' satisfy the given inequality.

Ans. (C)
_________________
Regards,

PKN

Rise above the storm, you will find the sunshine
Manager
Joined: 08 Oct 2015
Posts: 215
If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

31 Oct 2018, 09:52
2
How did you get 1 2 5 6 7 and 8 from those 2 intervals?

Posted from my mobile device
Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 950
WE: Supply Chain Management (Energy and Utilities)
Re: If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

31 Oct 2018, 09:57
1
rahulkashyap wrote:
How did you get 1 2 5 6 7 and 8 from those 2 intervals?

Posted from my mobile device

Hi rahulkashyap,
a) x≤2 and x is a +ve integer. hence x could be: 1, 2
b) 4<x<9 and x is a +ve integer. hence x could be: 5,6,7,8
_________________
Regards,

PKN

Rise above the storm, you will find the sunshine
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3410
If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

02 Nov 2018, 04:12
2

Solution

Given:
• We are given that x is a positive integer
• And, we are given an inequality, $$\frac{(x - 2)}{(x^2 – 13x + 36)} ≤ 0$$

To find:
• We need to find out the number of values of x that satisfies the given inequality

Approach and Working:
• If we observe the given inequality, we can see that the denominator is a quadratic expression.
• So, let’s try to factorise the quadratic expression, $$x^2 – 13x + 36$$
o $$x^2 – 13x + 36 = (x – 9) * (x – 4)$$

• Thus, the inequality can be written as, $$\frac{(x - 2)}{(x – 9)(x - 4)} ≤ 0$$
• Now, we need to multiply the numerator and denominator of LHS with (x – 9)*(x – 4)
o $$\frac{(x - 2) (x – 9)(x - 4)}{[(x – 9)(x - 4)]^2} ≤ 0$$
o The denominator of the above inequality is always greater than 0 and x ≠ 4 or 9, since the denominator cannot be equal to 0.
o So, the numerator must be ≤ 0, for $$\frac{(x - 2)(x – 9)(x - 4)}{[(x – 9)(x - 4)]^2} ≤ 0$$.
o We need to find the values of x, for which (x - 2) (x – 4)(x - 9) ≤ 0

Approach 1: Wavy-line method

• The zero points of the inequality are {2, 4, 9} and the wavy-line will be as follows:

• So, the expression will be equal to zero for x = {2, 4, 9}
o But, as we already know that, x cannot take the values 4 and 9
o Thus, x = {2}
• The expression will be negative in the regions, 4 < x < 9 and x < 2
o But we know that x is a positive integer, so, the expression will be negative for x = {1, 5, 6, 7, 8}
• Therefore, the given inequality satisfies for x = {1, 2, 5, 6, 7, 8}

Approach 2: Number-line method

The zero points of the inequality are {2, 4, 9} and highlighting these points on a number line, we get:

• Thus, (x - 2) (x – 4)(x - 9) will be ≤ 0 in two regions, x ≤ 2 and 4 ≤ x ≤ 9, but we already inferred that x ≠ {4 , 9}
• And we know that x is a positive integer
• Therefore, the given inequality satisfies for x = {1, 2, 5, 6, 7, 8}

Hence, the correct answer is option C.

_________________
VP
Joined: 09 Mar 2016
Posts: 1254
If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

02 Nov 2018, 06:22
EgmatQuantExpert wrote:

Solution

Given:
• We are given that x is a positive integer
• And, we are given an inequality, $$\frac{(x - 2)}{(x^2 – 13x + 36)} ≤ 0$$

To find:
• We need to find out the number of values of x that satisfies the given inequality

Approach and Working:
• If we observe the given inequality, we can see that the denominator is a quadratic expression.
• So, let’s try to factorise the quadratic expression, $$x^2 – 13x + 36$$
o $$x^2 – 13x + 36 = (x – 9) * (x – 4)$$

• Thus, the inequality can be written as, $$\frac{(x - 2)}{(x – 9)(x - 4)} ≤ 0$$
Now, we need to multiply the numerator and denominator of LHS with (x – 9)*(x – 4)
o $$\frac{(x - 2) (x – 9)(x - 4)}{[(x – 9)(x - 4)]^2} ≤ 0$$
o The denominator of the above inequality is always greater than 0 and x ≠ 4 or 9, since the denominator cannot be equal to 0.
o So, the numerator must be ≤ 0, for $$\frac{(x - 2)(x – 9)(x - 4)}{[(x – 9)(x - 4)]^2} ≤ 0$$.
o We need to find the values of x, for which (x - 2) (x – 4)(x - 9) ≤ 0

Approach 1: Wavy-line method

• The zero points of the inequality are {2, 4, 9} and the wavy-line will be as follows:

• So, the expression will be equal to zero for x = {2, 4, 9}
o But, as we already know that, x cannot take the values 4 and 9
o Thus, x = {2}
• The expression will be negative in the regions, 4 < x < 9 and x < 2
o But we know that x is a positive integer, so, the expression will be negative for x = {1, 5, 6, 7, 8}
• Therefore, the given inequality satisfies for x = {1, 2, 5, 6, 7, 8}

Approach 2: Number-line method

The zero points of the inequality are {2, 4, 9} and highlighting these points on a number line, we get:

• Thus, (x - 2) (x – 4)(x - 9) will be ≤ 0 in two regions, x ≤ 2 and 4 ≤ x ≤ 9, but we already inferred that x ≠ {4 , 9}
• And we know that x is a positive integer
• Therefore, the given inequality satisfies for x = {1, 2, 5, 6, 7, 8}

Hence, the correct answer is option C.

i would appreciate you could take time to answrr my questions below thank you!

Question 1. Why do we need to multiply the numerator and denominator of LHS with (x – 9)*(x – 4) in which cases do we need to do so ?

Question 2. if the denominator of the above inequality is always greater than 0 and x ≠ 4 or 9, since the denominator cannot be equal to 0. WHY do we remove whole denominator in the end, leave only this (x - 2) (x – 4)(x - 9) ≤ 0 ???

Question 3. And the last question is it gramatically paralel structure to say "Approach and Working" may be "Aprroach and Work" is better or "Approach first and then start Working" how about "approach while working and by the time you approach it will be solved"
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3249
Location: India
GPA: 3.12
If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

02 Nov 2018, 07:06
1
1
1
dave13 wrote:

i would appreciate you could take time to answrr my questions below thank you!

Question 1. Why do we need to multiply the numerator and denominator of LHS with (x – 9)*(x – 4) in which cases do we need to do so ?

Question 2. if the denominator of the above inequality is always greater than 0 and x ≠ 4 or 9, since the denominator cannot be equal to 0. WHY do we remove whole denominator in the end, leave only this (x - 2) (x – 4)(x - 9) ≤ 0 ???

Question 3. And the last question is it gramatically paralel structure to say "Approach and Working" may be "Aprroach and Work" is better or "Approach first and then start Working" how about "approach while working and by the time you approach it will be solved"

Hey dave13

The answer to question 1 is that they do so to make the denominator a square.
Remember a square of a number/expression can never be negative. Now, that
we have that in order, for the overall expression to be neagtive or equal to zero,
the numerator must be negative or have a value of zero at maximum.

That's the reason why they remove the denominator and keep only the numerator

As for your thid question, I am sure generis will have an answer. Unfortunately,
I am not qualified to make that judgement
_________________
You've got what it takes, but it will take everything you've got
Senior PS Moderator
Status: It always seems impossible until it's done.
Joined: 16 Sep 2016
Posts: 719
GMAT 1: 740 Q50 V40
GMAT 2: 770 Q51 V42
Re: If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

02 Nov 2018, 07:38
1
dave13,

I'd like to add that this is a common trick in inequality questions, to make perfect squares and then "ignore them" as the perfect squares of real numbers are greater than or equal to zero.

However, as an advanced step you can go ahead and skip that part. Treat the numerator and denominator the same and use the wavy approach. ( While multiplying by same factor in both num and den we are doing this itself!)

To add a little to q2 raised by you above, -ve * +ve is -ve and +ve * +ve is +ve... Which means multiplying by positive entity does not change the sign... Hence we can ignore the perfect square part in denominator.

Q3: no clue what you're talking about!

Let me know if you wanna followup.

Best,

Posted from my mobile device
_________________
Regards,

“Do. Or do not. There is no try.” - Yoda (The Empire Strikes Back)
CEO
Joined: 03 Jun 2019
Posts: 3230
Location: India
GMAT 1: 690 Q50 V34
WE: Engineering (Transportation)
Re: If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

28 Mar 2020, 09:25
EgmatQuantExpert wrote:
If x is a positive integer, how many values of x satisfy the inequality, $$\frac{(x - 2)}{(x^2 – 13x + 36)} ≤ 0$$?

A. 4
B. 5
C. 6
D. 7
E. Cannot be determined

Asked: If x is a positive integer, how many values of x satisfy the inequality, $$\frac{(x - 2)}{(x^2 – 13x + 36)} ≤ 0$$?

(x-2)/(x-4)(x-9) <=0

Possible values of x = {1,2,5,6,7,8}

IMO C
_________________
Kinshook Chaturvedi
Email: kinshook.chaturvedi@gmail.com
Intern
Joined: 04 Apr 2019
Posts: 9
Re: If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

29 Mar 2020, 05:47
Why can't 0 be one of the integer.

Posted from my mobile device
Intern
Joined: 23 Mar 2020
Posts: 1
Re: If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

12 Apr 2020, 11:09
rahulkashyap wrote:
How did you get 1 2 5 6 7 and 8 from those 2 intervals?

Posted from my mobile device

3 is also satisfying the equation
Manager
Joined: 24 Jan 2013
Posts: 95
Location: India
GMAT 1: 660 Q49 V31
Re: If x is a positive integer, how many values of x satisfy the..........  [#permalink]

### Show Tags

25 Apr 2020, 06:57
EgmatQuantExpert wrote:
If x is a positive integer, how many values of x satisfy the inequality, $$\frac{(x - 2)}{(x^2 – 13x + 36)} ≤ 0$$?

A. 4
B. 5
C. 6
D. 7
E. Cannot be determined

Attachments

File comment: You can use this method of plotting transition points on the number line.

This is fast method and with practice one can get used to it.

6226da61-516a-4a8a-b3d5-c724a9240d3f.jpg [ 81.43 KiB | Viewed 537 times ]

_________________
Re: If x is a positive integer, how many values of x satisfy the..........   [#permalink] 25 Apr 2020, 06:57